On super Pl\"{u}cker embedding and cluster algebras

Abstract

We define a super analog of the classical Pl\"{u}cker embedding of the Grassmannian into a projective space. One of the difficulties of the problem is rooted in the fact that super exterior powers $\Lambda^{r|s}(V)$ are not a simple generalization from the completely even case (this works only for $r|0$ when it is possible to use $\Lambda^r(V)$). To construct the embedding we need to non-trivially combine a super vector space $V$ and its parity-reversion $\Pi V$. Our "super Pl\"{u}cker map" takes the Grassmann supermanifold $G_{r|s}(V)$ to a "weighted projective space" $P\left(\Lambda^{r|s}(V)\oplus \Lambda^{s|r}(\Pi V)\right)$ with weights $+1,-1$. A simpler map $G_{r|0}(V)\to P(\Lambda^r(V))$ works for the case $s=0$. We construct a super analog of Pl\"{u}cker coordinates, prove that our map is an embedding, and obtain "super Pl\"{u}cker relations". We analyze another type of relations (due to Khudaverdian) and show their equivalence with the super Pl\"{u}cker relations for $r|s=2|0$. We discuss application to much sought-after super cluster algebras and construct a super cluster structure for $G_2(\mathbb{R}^{4|1})$ and $G_2(\mathbb{R}^{5|1})$.